Open Problems in the Spectral Analysis of Evolutionary Dynamics

Abstract

For broad classes of selection and genetic operators,the dynamics of evolution can be completelycharacterised by the spectra of the operators thatdefine the dynamics, in both infinite and finitepopulations. These classes include generalisedmutation, frequency-independent selection, uniparentalinheritance. Several open questions exist regardingthese spectra: 1. For a given fitness function, whatgenetic operators and operator intensities are optimalfor finding the fittest genotype? The concept of rapidfirst hitting time, an analog of Sinclair's rapidlymixing Markov chains, is examined.2. What is the relationship between the spectra ofdeterministic infinite population models, and thespectra of the Markov processes derived from them inthe case of finite populations?3. Karlin proved a fundamental relationship betweenselection, rates of transformation under geneticoperators, and the consequent asymptotic mean fitnessof the population. Developed to analyse the stabilityof polymorphisms in subdivided populations, the theoremhas been applied to unify the reduction principle forself-adaptation, and has other applications as well.Many other problems could be solved if it weregeneralised to account for the interaction of differentgenetic operators. Can Karlin's theorem on operatorintensity be extended to account for mixed geneticoperators?